1. (3-2) Algorithms for Whole-Number Addition and Subtraction
1.1. Algorithms- A systematic procedure used to accomplish an operation. I. Addition Algorithms a. To help students understand algorithms, we should start with manipulatives. Children can touch, move around, and be led to developing their own algorithms (a procedure to accomplish an operation.) (See Figure 3-12 on pg. 128) b. After working with manipulatives, then move to paper/pencil operations. (See pg. 128 and School Book Page on pg. 129) c. Regroup or trade problems are then used to describe carrying. 1 Example: 37 + 28 65 d. Lattice Algorithm for Addition Example: 3 5 6 7 + 5 6 7 8 0/1/1/1 /8/1/3/5 9 2 4 5
1.2. II. Subtraction Algorithms a. Use base-ten blocks to provide a concrete model for subtraction as we did in addition. b. The concept of remove or take away is used. c. Then paper/pencil algorithms are introduced.
1.3. III. Equal-Addition Algorithm a. Based on the fact that the difference between two numbers does not change if we add the same amount to both numbers. Example: 255 > 255 + 7 > 262 > 262 + 30 > 292 - 163 163 + 7 -170 -(170 + 30) - 200 92
1.4. IV. Understanding Addition and Subtraction in Bases Other Than Ten
2. Test two
2.1. (3-1) Addition of Whole Numbers
2.1.1. Addition of Whole Numbers Set Model Example: Set A n(A)= {a,b,c,d} Set B n(B)={e,f,g} n(A) + n(B)= {a,b,c,d,e,f,g} =4+3=7= n(A U B) The numbers a + b are the ADDENDS And a + b is the SUM.
2.1.1.1. I. Number Line (Measurement) Model Solve 4+3 using a number line. ALWAYS start at ZERO with your first number!! <--I---I---I---I---I---I---I---I---I---I---I---I---> 0 1 2 3 4 5 6 7 8 9 10 11 II. Ordering Whole Numbers >Greater than 3>n <--I---I---I---I---I---I---I---I---I---I---I---I---> 0 1 2 3 4 5 6 7 8 9 10 11 < Less than 2<n <--I---I---I---I---I---I---I---I---I---I---I---I---> 0 1 2 3 4 5 6 7 8 9 10 11 = Equal 2=n >Greater than or equal to 4>n <--I---I---I---I---I---I---I---I---I---I---I---I---> 0 1 2 3 4 5 6 7 8 9 10 11 <Less than or equal to 3< n <--I---I---I---I---I---I---I---I---I---I---I---I---> 0 1 2 3 4 5 6 7 8 9 10 11 III. Addition Properties A. Closure Property- If a and b are whole numbers, then a + b is a whole number. Example: 5+2=7 S= {0,7,14,21…} is closure because 0+7=7 , 7+14=21, etc. S={0} is closure because 0+0=0 S={7,10, 12, 13…} is NOT closure because 7+10=17 (17 is not a number in set) Please note: C = { x | x Є N where x < 5} which means Set C = the set of – all elements x- such as – x is a natural number- where x is less than 5 C ={x | x W, x>14} which means Set C= the set of- all elements x – such as - x is a whole number- where x is greater than 14 (If any whole number greater than 14 is added to another whole number greater than 14, the sum is great than 14.) This is a closed property. B. Commutative Property- a+b=b+a Example= 6+7=7+6 C. Associative Property- (a+b)+c= a+(b+c) Example= (3+4)+5= 3+(4+5) - Also known as the grouping property D. Identity Property- a+0=a Any number plus 0 is that number. Example= 2+0=2
2.1.1.1.1. Mastering Basic Addition Facts (Single digit plus a single digit) A. Counting On- 4+2= 4,5,6 also 3+3= 3, 4, 5,6 B. Doubles- Master doubles, then double plus 1, double plus 2 etc. Example= Know 3+3=6, then be able to calculate 3+4 by 3+3=6 plus one more= 7 C. Making 10 (and then add any leftovers) Example: 8+5=(8+2)+3=13 D. Counting Back- Used when one number is 1 or 2 less than 10. Example= 9+7. 9 is one less than 10 which equals (10+7)-1= 16 or 8+7= (10+7)-2=15 *Inductive Reasoning: Method of making generalizations based on observations and patterns. **Fact Families for addition: 3+8=11 8+3=11 11-3=8 11-8=3
2.1.2. Subtraction of Numbers Inverse Operations- operations that “undo” each other. Subtraction is the inverse of addition. Model with: A. Take-Away Model- You have 8 blocks, take away 3 blocks= 8-3=5 blocks left B. Missing Addend Model- Relates addition and subtraction 3 + ___ = 8 Put in 3 blocks plus ???= 8 8-3= 5 3+ ?= 8 (Start of algebraic thinking.) Can also use a number line n <--I---I---I---I---I---I---I---I---I---I---I-> 0 1 2 3 4 5 6 7 8 9 10 Use fact families Cashiers often use missing addend (Movie costs $8.00 you pay $10.00, thus 8+2=10. C. Comparison Model- Juan’s 8 blocks Susan’s 3 blocks D. Number Line Model 5-3=2 <--I---I---I---I---I---I---I---I---I---I---I---I---> 0 1 2 3 4 5 6 7 8 9 10 11 E. Properties of Subtraction- Which addition properties work for subtraction a. Closure- {1,3,5,7,…} (3-5=-2) NO- Answer is not a WHOLE number b. Associative (a-b)-c=a-(b-c) YES c. Commutative- a-b=b-a NO d. Identity- a-0=a YES However: 0-a=0 is not true.
2.2. (3-3) Multiplication and Division of Whole Numbers
2.2.1. I. Multiplication of Whole Numbers • Repeated-Addition Model • We can use addition to put equal groups of numbers together to use multiplication. 3 + 3 + 3 + 3 = 12 (four groups of 3’s) • Can be shown by number lines and arrays. (See pg. 143) • The constant feature (+) on a calculator can help relate multiplication to addition. Example: + 3 = = = = 12
2.2.2. • Cartesian-Product Model Use of a tree diagram to solve multiplication problems (See pg. 146) *Be aware of how multiplication is modeled: *A X B, A(B), A B where A and B are the factors and A X B is the product.
2.2.3. II. Properties of Whole Number Multiplication A. Closure property of multiplication of whole numbers- The set of whole numbers is closed under multiplication. That is, if we multiply any two whole numbers, the result is a unique whole number. B. Commutative property of multiplication of whole numbers- For whole numbers a and b, a X b = b X a. C. Associative property of multiplication of whole numbers- For whole number a, b, and c, (a X b) X c = a X (b X c) D. Identity property of multiplication of whole numbers- There is a unique whole number 1 such that for any whole number a, a X 1 = A = 1 X a E. Zero multiplication property of whole numbers- For any whole number a, a X 0 = 0 = 0 X a F. Distributive property of multiplication over addition and subtraction- For any whole numbers a, b, and c, a(b+c)= ab + ac and a(b-c) = ab – ac Example of how distributive property works: 7 X 13 = 7 X (10 + 3) = (7 X 10) + (7 X 3) = 70 + 21 + 91 **The purpose of distributive property is the enable the student to solve the problem using mental math.
2.2.4. III. Division of Whole Numbers a. Set (Partition) Model- Set up a model of the total number of items in the problem then partition them into sets. Example: 18 cookies divided by 3 would be 3 sets of 6 cookies. b. Missing-Factor Model- Using multiplication, the number of groups times the unknown variable is equal to the total. Example: 3 X c = 18 By using multiplication, we know that 3 X 6 equals 18, thus c = 6. **Key vocabulary: In the division problem a divided by b equals c. A is the dividend, b is the divisor, and c is the quotient. Note that a divided by b can be written as a = c b c. Repeated subtraction model- Example: 18 divided by 6 could be shown as 18- 6 = 12 – 6 = 6 – 6 = 0 or 18 - 6 -6 -6 = 0 IV. The Division Algorithm • Given any whole numbers a and b with b = 0, there exist unique whole numbers q (quotient) and r (remainder) such as a=bq + r with 0 < r < b • When a is “divided” by b and the remainder is 0, we say that a is divisible by b or that b is a divisor of a or that b divides a. V. Relating Multiplication and Division as Inverse Operations *Division is the inverse of multiplication. *Division with a remainder of 0 and multiplication are related. *Note—is division closed, commutative, associative, and/or identity property? VI. Division by 0 or 1 ( bottom of pg. 154 and see School Book Page on pg. 155) • n divided by 0 is undefined (there is no answer to the equivalent multiplication problem.) • 0 divided by n = 0 • 0 divided by 0 is undefined also. VII. Order of Operations When solving equations, students have difficulties involving the order of arithmetic operations. A particular order is required and can be memorized by Please Excuse My Dear Aunt Sally. The order is: 1. Parenthesis 2. Exponents 3. Multiplication or whichever one comes first left to right Division 4. Addition or whichever one comes first left to right Subtraction
2.3. (4-3) Functions
2.3.1. A FUNCTION is a relationship that assigns exactly one output value for each input value. A FUNCTION from set A to Set B is a correspondence from A to B in which each element of A is paired with one, and only one, element of B.
2.3.2. Ways to represent functions: A. Functions as Rules What’s the Rule? n x 3 *Each element of A is paired with exactly one element of B. You Teacher 1 3 0 0 4 12 10 30
2.3.3. B. Functions as Machines What goes in >>Input What comes out >>Output (x) (function) < f (x) = (d) Read “f of x” When talking about distance (d) and time (t) ----------------------------------------------------------------------------- Input: time (h) 1 2 3 Output: distance (mi) 55 110 165 d= 55 x t 4 x f(x) Add 3 0 3 f(4)=7 1 4 3 6 4 7 6 9
2.3.4. Functions as Equations f (0) = 0+3=3 f (1) = 1+3=4 f (3) = 3+3=6
2.3.5. D. Functions as Arrow Diagrams Used to examine whether a correspondence represents a function. Domain Range 0 1 1 4 Yes, a function. 2 7 3 10 1 2 2 4 No, since element 1 is paired with 2 & 4. 3 John Brown Mike Smith Yes, since there is one and only one arrow leaving each Joan Doe element in A. It does not matter that an element of set Sue B, Brown, has two arrows pointing to it. The range is (Brown, Smith, Doe.)
2.3.6. II. Integer Division The quotient of two negative integers, is a positive integer and the quotient of a positive and negative integer, is negative. **Do Example 5-13 for further practice. III. Order of Operations on Integers Use the order of operations: parenthesis, exponents, multiply/divide left to right/ add/subtract left to right. (2-5)4 + 1 2 – 3 x 4 + 5 x 2 – 1 + 5 (-3)4 + 1 2 – 12 + 10 – 1 + 5 -12 + 1 -10 + 10 – 1 + 5 -11 0 – 1 + 5 -1 + 5 4 IV. Order of Integers x + 3 < -2 x + 3 + -3 < -2 + -3 x < -5 ( -6, -7, -8, . . . .) -x – 3 < 5 -x – 3 + 3 < 5 + 3 -x < 8 -(-x) > -8 (look at Theorem 5-11 – pg. 277) x > -8 (-7, -6, -5,. . .) -3x > -3(-2) -3x > 6 5 + -3x > 5 + 6 5 – 3x > 5 + 6 5 – 3x > 11 ; that is all integers (11, 12, 13, 14, …..)
2.3.6.1. MAT 156 Session 17- (5-3) Divisibility Mathematics for Elementary Teachers I
2.3.6.1.1. If b|a, then b is a factor, or divisor, of a, and a is a multiple of b. We also assume that a and b are also integers and b = 0
2.3.6.1.2. Do not confuse b|a and b/a, which the latter is interpreted as b divided by a.
2.3.6.1.3. Examples: -3|12 is true because 12 = -4 (-3)
2.3.6.1.4. 0|2 is false because there is no integer c such that 2 = c x 0
2.3.7. E. Functions as Tables and Ordered Pairs 0 1 (0,1) (1,3), (2,5), (3,7), (4,9) 1 3 2 5 3 7 4 9 Functions? a) (1,2) (1,3) (2,3) (3,4) No, 1 input used twice b) (1,2) (2,3) (3,4), (4,5) Yes c) (1,0) (2,0) (3,0), (4,4) Yes 1 0 2 3 4 4
2.3.8. F. Functions as Graphs Horizontal- inputs Vertical- outputs
2.3.9. G. Sequences as Functions II. Relations A relation from Set A to set B is a correspondence between elements of A and element of B, but unlike functions, do not require that each element of A be paired with one, and only one, element of B. • Every function is a relation, but not every relation is a function.
3. C. Number-Line Model- see pgs. 271-272 1. When moving on a number line, moving to the left means moving in the negative direction, and traveling to the right means moving in the positive direction. 2. Time in the future is denoted by a positive value, and time in the past is denoted by a negative value. 3. Note that 3 x -2 means three groups of -2 each or 3 x -2 = -6 D. Properties of Integer Multiplication 1. Closure property of multiplication of integers – ab is a unique integer 2. Commutative property of multiplication of integers- ab = ba 3. Associative property of multiplication of integers = (ab)c = a(bc) 4. Multiplicative identity property – 1 x a = a = a x 1 5. Distributive properties of multiplication over addition of integers =a(b+c) = ab +ac and (b +c)a= ba + ca 6. Zero multiplication property of integers – a x 0 = 0 = 0 x a Page 2 7. Additive Inverse- (2x3) is –(2x3) thus (2x3) + (-2)(3) = 0 • Note- (-2)3 = -(2x3) thus for every integer a, (-1)a= -a • (-a)b= -(ab) and (-a)(-b) = ab • a(b-c) = ab-ac and (b-c)a= ba-ca • Do example 5-10, 5-11, and 5-12 for further practice.
4. I. Prime Factorization 1. Composite numbers can be expressed as products of two or more whole numbers greater than 1. For example, 18 = 2x9, 3x6, 2x3x3. Each expression is a factorization. 2. Prime factorization- a factorization containing only prime numbers. 3. Factor tree- 24 4 x 6 2 x 2 x 3 x 2 Answer: 23 x 3 4. Ladder model – see page 303 2 12 2 6 3 3 1 5. Number of Divisors How many positive divisors does 24 have? Note that the question asks for the number of divisors, not just prime divisors. Group divisors as follows: 1 , 2, 3, 4, 6, 8, 12, 24 Also, think of the number of positive divisors of 24, consider the prime factorization 24 = 23 x 3. The positive divisors of 23 are 20, 21, 22, 23. The positive divisors of 3 are 30 and 31. We know that 23 has (3+1), or 4, divisors and 31 has (1 + 1), or 2 divisors. Because each divisor of 24 is the product of a divisor of 23 and 31, the we use the Fundamental Counting Principle (pg. 82) to conclude that 24 has 4 x 2, or 8 positive divisors. 6. Sieve of Eratosthenes – Method for identifying prime numbers- pg. 309
4.1. MAT 156 Session 19- (5-5) Greatest Common Divisor and Least Common Multiple
4.1.1. I. Greatest Common Divisor (Factor)- GCF The greatest common divisor of two natural numbers a and b is the greatest natural number that divides both a and b. a. Colored Rods Method – see page 315 b. The Intersection of Sets Method- List all members of the set of positive divisors of the two integers, then find the set of common divisors, and finally, pick the greatest element in that set. Set of 20- 1,2,4,5,10,20 Set of 32- 1,2,4,8,16,32 Thus the common positive divisors or factors of 20 and 32 are 1,2,4. Four is the greatest, so 4 is your greatest common divisor (factor). You can also make a Venn diagram of the factors to find the GCD. (Pg. 316) c. Prime Factorization Method- 180 = 2 x 2 x 3 x 3 x 5 = (22 x 3) 3 x 5 168 = 2 x 2 x 2 x 3 x 7 – (22 x 3)2 x 7 Thus the common prime factorization of 180 and 168 is 22 x 3= 12 D. Calculator Method – pg. 317 E. “Ladder Method” or Division by Primes Method To find the GCD of 75 and 100 5 75, 100 5 15, 20 3, 4 II. Least Common Multiple The least common multiple (LCM) of a and b is the least natural number that is simultaneously a multiple of a and multiple of b. a. Number-Line Method ---l---l---l---l---l---l---l---l---l---l-----l-----l-----l-----l-----l-----l----l----l-----l-----l-----l- 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 b. Colored Rods Method- pg. 322 Page 2 c. Intersection of Sets Method First find the multiples of both the first and second numbers, then find the set of all common multiples of both numbers, and finally pick the least element in that set. 8- 8, 16, 24, 32, 40, 48, 56, 64, 72, . . . 12- 12, 24, 36, 48, 60, 72, 84, 96, 108, . . . Common multiples are 24, 48, 72, but the LEAST common multiple is 24 written LCM(8,12) = 24 d. Prime Factorization Method To find the LCM of two natural numbers, first find the prime factorization of each number. Then take each of the primes that are factors of either of the given numbers. The LCM is the product of these primes, each raised to the greatest power of the prime that occurs in either of the prime factorizations. Example: 12= 22 x 3 108 = 22 x 33 120 = 23 x 3 x 5 LCM(12,108.120) = 23 x 33 x 5 = 1080 e. “Ladder Method” or The Division by Primes Method 2 12,75,120 2 6, 75, 60 2 3, 75, 30 2 3, 75, 15 3 3, 75, 5 5 1, 25, 1 1, 5, 1 Thus, LCM (12, 75, 120) = 2x2x2x2x3x5x1x5x1= 600
5. IV. Cardinal Numbers- Rd. page 84. The cardinal number of a set denotes the number of elements in the set. Example: The sets, (a,b), (p,q), (x,y), (b,a), (*,#) are equivalent to one another and share the property of “twoness”; that is these sets have the same cardinal number, namely, 2.
6. II. Arithmetic Sequences is one in which each successive term from the second term on is obtained from the previous term by addition or subtraction of a fixed number. A. Example: 3, 6, 9, 12…. Where “add 3” is the sequence. B. Recursive pattern- After one or more consecutive terms are given to start, each successive term of the sequence is obtained from the previous term.
7. Chapter 1-1- Mathematics and Problem Solving Essential Question: Why is teaching problem solving an important part of mathematics?
7.1. Four-Step Problem Solving Process: 1. Understand the problem 2. Devising a plan 3. Carrying out the plan 4. Looking back (Check!)
8. Chapter 1-2- Explorations with Patterns Essential Question: Why is understanding numerical and geometric patterns important in mathematics and the real world?
8.1. I. Inductive Reasoning- Method for making generalizations based on observations and patterns. This type of reasoning leads to: A. Conjecture- Statement thought to be true, but not yet proven to be true or false. Such as zero squared= 0, one squared=one….But when we find it to be false… B. Counterexample- proves the conjecture is false in general such as two squared=four.
8.2. III. Fibonacci Sequence- This sequence is named after the Italian Leonardo de Pisa better known as Fibonacci. This sequence is NOT arithmetic as there is no fixed difference. A. An example of this sequence is 1,1,2,3,5,8,13,21,34,55,89,144…….
8.3. IV. Geometric Sequences- Each successive term of a geometric sequence is obtained from its predecessor by multiplying by a fixed nonzero number, the ratio. A. Example: A child has 2 biological parents, 4 grandparents, 8 great grandparents, 16 great-great grandparents, and so on. In this case, the first term and the ratio are 2.
9. Chapter 2-1 Numeration Systems Essential Question: Why is it necessary for students to understand various ancient number systems and compare them to the system of numbers that we use today in the United States?
9.1. Numerals- The written symbols for digits.
9.1.1. I. Hindu-Arabic Numeration System a. All numerals are constructed from the 10 digits- 0,1,2,3,4,5,6,7,8,and 9. b. Place value is based on powers of 10, the number base of the system. c. Expanded form of a number uses the place value: 2,345= 2000+300+40+5 **Work with base-ten blocks- unit- 1 block/ longs- 10 units/ flat- 100 units/ block- 1000 units d. Solve this problem using the base ten blocks: What is the fewest number of pieces you can receive in a fair exchange of 1 flat, 5 longs, and 16 units?________________________ Now do it on paper: 11 flats 17 longs 16 units +1 long 6 units TRADE 11 flats 18 longs 6 units
9.1.2. II. Tally Numeration System – uses single strokes, or tally marks, to represent each object that was counted: for example, the first five counting numbers are: I, II, III, IIII, IIIII
9.1.3. III. Egyptian Numeration System- See page 66 for these IV. Babylonian Numeration System- See page 67 V. Mayan Numeration System- See page 68
9.1.4. VI. Roman Numeration System- Uses the following symbols with meanings of: I =1 L=50 Examples of equations: V= 5 C=100 IV= (5-1) 4 X = 10 D=500 IX= (10-1) 9 M=1000 XL= (50-10) 40 Do: CCLIV ________________________
9.1.5. VII. Other Number Base Systems We currently use the ten number base system. If there is no base noted, then the number is in the ten number base. Other number base systems are used. One example is the Base Five. Look at page 70 for examples. Now let’s figure out other base five numbers. 14five = 1 x 5 + 4 = 9 124five = 1 x 52 + 2 x 5 + 4= 25 + 10 + 4 = 39 1030five= 1 x 53 + 0 x 52 + 3 x 5 + 0 = 125 + 0 + 15 + 0 = 140
10. Chapter 2-2 & 2-3 Sets Essential Question: How are sets, and relations between sets, a basis to teach children the concept of whole numbers and many other necessary operations such as addition, subtraction, and multiplication of whole numbers?
10.1. Language of Sets- A set is understood to be any collection of objects. Individual objects in a set are elements or members of the set. Example: The set A of lowercase letters can be written A=(a,b,c,d,e,f,...) The order in which the elements are written makes no difference, and each element is listed only once.
10.2. II. One-to-One Correspondence- If the elements of sets P and S can be paired so that for each element of P there is exactly one element of S and for each element of S there is exactly one element of P, then the two sets P and S are said to be in one to one correspondence. Other possible one-to-one correspondences exist: Example: Tomas 1 1. Tomas—1 1. Tomas--2 Dick 2 2. Dick—3 2. Dick--1 Mari 3 3. Mari-2 3. Mari--3
10.3. III. Equivalent Sets-Two sets A and B are equivalent, written A ~B if and only if there exists a one-to-one correspondence between the sets. **The term equivalent should not be confused with equal. Example: A= (p,q,r,s), B= (a,b,c) C= (x,y,z) and D= (b,a,c) Sets A and B are not equivalent and not equal. Sets B and C are equivalent, but not equal.
10.4. V. Sets- Universal set or the universe, denoted U, is the set that contains all elements being considered in a given discussion. See Figure 2-12 on page 85 as an example. -- A Venn diagram is used to illustrate sets. Page 85
10.5. VI. Subsets- B is a subset of A, if, and only if, every element of B is an element of A. Example: A= (1,2,3,4,5,6) and B=(2,4,6) All of the elements of B are contained in set A and we say that B is a subset of A.
10.6. VII. Set Intersection- The intersection of two sets, A and B, is the set of all elements common to both A and B. Example: A= (1,2,3,4) and B= (3,4,5,6) Thus, A intersects with B-= ((3,4)
10.7. VIII. Set Union-The union of two sets A and B, is the set of all elements in A or in B. Example: A= (1,2,3,4), B= (3,4,5,6) The set union would be (1,2,3,4,5,6)
10.8. IX. Set Difference- (Also called the relative complement.) The complement of A relative to B, is the set of all elements in B that are not in A. Example: A= (d,e,f), B= (a,b,c,d,e,f), and C= (a,b,c), thus; the set difference would be (a,b,c).
10.9. X. Properties of Set Operations- Associative Property Distributive Property
10.10. XI. Cartesian Products- For any sets A and B, the Cartesian produce of A and B, written A X B, (read as “A cross B”) is the set of all ordered pairs such that the first component of each pair is an element of A and the second component of each is an element of B. Example: Suppose a person has three pairs of pants, P=(blue, white, green) and two shirts, S=(blue,red). There are 3 X 2, or 6 possible different pant-and-shirt pairs. See Figure 2-23 on page 100.
11. Test one
12. Test three
12.1. Integers and the Operations of Addition and Subtractions Integers- positive or negative whole numbers including zero. The negative integers are opposites of the positive integers. For example, the opposite of 5 is -5. Similarly, the positive integers are the opposite of the negative integers. Because the opposite of 4 is denoted -4, the opposite of -4 can be denoted -(-4), which equals 4. The opposite of 0 is 0. Note: -x is read “the opposite of x” not “minus x” or “negative x.”
12.2. Integer Addition A. Chip Model for addition- Black chips are used to represent positive integers and negative integers are red. (We have two sided chips- yellow (positive) and red (negative). B. Charged-Field Model - Similar to chips but use the positive (+) and (-) charges. C. Number Line Model- Always start at zero (0) and move line to first number, then go to second number. D. Pattern Model- Addition of integers can also be motived by using patterns of addition of whole numbers. (pg. 254) E. Absolute Value – The distance between the point corresponding to an integer and 0 is the absolute value of the integer. Thus, the absolute value of both 4 and -4, written l4l and l-4l = 4. Notice that if x > 0, the lxl = x and if x < 0, then –x is positive.
12.3. Properties of Integer Addition a. Closure property of addition of integers b. Commutative property of addition of integers c. Associative property of addition of integers d. Identity element of addition of integers- 0 is the unique integer such that, for all integers a, 0 + a = a = a + 0 Uniqueness of the Additive Inverse- For every integer a, there exists a unique integer –a, the additive inverse of a, such that a + -a = 0 = -a + a. Also, -(-a)= a and –a + -b = -(a+b)
12.4. MAT 156 Session 16- (5-2) Multiplication and Division of Integers
12.4.1. Integer Subtraction A. Chip Model for Subtraction- Same as addition, but notice when you have 3 - -2 you must first put in 3 black chips, then put in 2 black and red chips, then subtract 2 red chips to make the equation true. B. Charged Field Model for Subtraction C. Number Line Model for Subtraction D. Pattern Model for Subtraction E. Subtraction Using the Missing Addend Approach- 5-3 = n if and only if 5 = 3 + n (pg. 261) F. Subtraction Using Adding the Opposite Approach- a-b= a + -b Example: 2-8 = 2 + -8 = -6 -(b-c) = -(b+ -c)= -b + - (-c) = -b + c We often teach subtraction of integers with the saying, “keep-change-change” G. Properties of Subtraction Cannot do commutative nor associative with subtraction
12.4.2. **Remember the following: Multiply or divide a positive and a positive the answer is a positive. Multiply or divide a positive and a negative the answer is negative.
12.4.2.1. I. Multiplication of Integers A. Patterns Model for Multiplication of Integers 3 X -2 = -2 x -2 x -2 = -6 3(-2)= -6 2(-2)= -4 1(-2)=-2 0(-2)=0 -1(-2)=2 -2(-2)=4 B. Chip Model and Charged-Field Model for Multiplication- see pgs. 270-271 Please note: To find (-3)(-2) using the chip model, we interpret the sign as follows: -3 is taken to mean “remove 3 groups of”; -2 is taken to mean “ 2 red chips.” Start with a value of 0 that includes at least 6 red chips and 6 black chips. When we remove 6 red chips, we are left with 6 black chips. The result is a positive 6.
13. Test 4 Chapter 6 and 7
13.1. (6-1) The Set of Rational Numbers
13.1.1. I. In the rational number a/b, a is the numerator and b is the denominator. It can be represented as a/b or as “a divided by b”.
13.1.1.1. Uses of Rational Numbers: 1. Division problem or solution to a multiplication problem such as 2x = 3 is 3/2. 2. Partition, or part, of a whole such as Joe ate ½ of the pizza for dinner. 3. Ratio such as the ratio of girls to boys in the class was ten to twelve. 4. Probability such as when you toss a fair coin, the probability of getting heads is ½.
13.1.1.1.1. • Research has found teaching with the area model (part of a whole) is preferred over the set model (group of colored chips) See examples on page 343. When considering parts-to-whole model, we must consider the following: a. The whole being considered. b. The number b of equal-size parts into which the whole is divided. c. The number a of parts of the whole that we are selecting. • Rational number can be represented on a number line. See page 345 • Proper Fraction- A fraction a/b, where 0 < a < b or the numerator is smaller than the denominator. (1/2) • Improper Fraction- If a/b where a > b > 0 or the numerator is greater than the denominator. (3/2)
13.1.2. II. Equivalent or Equal Fractions A. Equivalent fraction are numbers that represent the same point on a number line. Examples are 1/3, 2/6, 3/9, 4/12, 5/15 . . . . B. The value of the fraction does not change if its numerator and denominator are multiplied by the same nonzero integer. C. Fundamental Law of Fractions. Let a/b be any fraction and n a nonzero integer. Then, a/b = an/bn. D. Is 7/-15 = -7/15? Yes because 7/-15 x -1/-1 = -7/15 E. Equivalent fractions can be found from dividing n/n into a fraction such as 12/42 = 2/7 x 6/6 = 2/7
13.1.3. III. Simplifying Fractions A. A rational number a/b is in simplest form if b > 0 and GCD (a,b) = 1; that is , if a and b have no common factor greater than 1, and b > 0. B. To write a fraction a/b in simplest form; that is in lowest terms, we divide both a and by the GCD. C. Example of simplifying a fraction: 60/120 divided 10/10 = 6/21 divided by 3/3 = 2/7 D. Let’s try a more difficult problem: 28ab2 = 2(14ab2) = 2 42a2b2 3a(14ab2) 3a
13.1.4. IV. Equality of Fractions - Can be shown three ways: A. Simply both fractions to the same simplest forms: 10/35 = 5x2 = 2/7 5x7 B. Rewrite both fractions with the same least common denominator. Since LCM (42, 35) = 210 then, 12/42 = 60/210 and 10/35 =60/210 C. Rewrite both fractions with a common denominator (not necessarily the least). Example: 12/42 = 420/1470 and 10/35 = 420/1470 hence: 12/42 = 10/35 Two fractions a/b and c/d are equal if, and only if, ad=bc
13.1.5. V. Ordering Rational Numbers A. Start by comparing fractions with like denominators such as 7/8 > 5/8. B. With unlike denominators, best to start with fraction strips or bars to compare the fractions visually. See page 354 C. Comparing fractions with unlike denominators can be accomplished by rewriting the fractions with the same positive denominator. Compare ¾ and 9/16. Find a common denominator (16), then change ¾ to 12/16 and compare to 9/16, thus; ¾ >9/16 D. Comparing rational numbers can be accomplished by estimating on a number line for the number closest to 0, ½, 1. Also you can look at the denominators to estimate the order.
13.1.6. VI. Denseness of Rational Numbers A. Given two different rational numbers a/b and c/d, there is another rational number between these two numbers. Example: ½ and 2/3 = 3/6 and 4/6 , because there is not number between 3 and 4, we next find two fractions equal, respectively, to ½ and 2/3. So ½ = 6/12 and 2/3 = 8/12, so 7/12 is between the two fractions.
13.2. (7-1 & 7-2) Introduction and Operations on Decimals
13.2.1. Decimals are most familiar when dealing with money. $125.67 Decimals places: 0 . 2 3 4 5 6 Ones. Tens Hundredths Thousandths Ten-Thousandths Hundred Thousandths 1/10 1/100 1/1000 1/10,000 1/100,000 5.4 = 5 + 0.4 = 5 + 4/10 = 50/10 + 4/10 = 54/10 Decimals can be written in expanded form; 12.618 = 1 ∙ 101 + 2 ∙ 100 + 6 ∙ 10-1 + 1 ∙ 10-2 + 8 ∙ 10-3 Convert each to decimals: 25/10 = 2 ∙ 10 + 5 = 2 ∙ 10 + 5/10 = 2 + 5/10 = 2.5 10 10
13.2.2. Terminating Decimals- decimals that can be written with only a finite number of places to the right of the decimal point .
13.2.3. Ordering Terminating Decimals – Change decimal by adding place value such 0.36 and 0.9 change to 0.36 and 0.90 to ease in ordering decimals. Line up decimals such as: 1. Line up the numbers by place value. 2. Start at left and find the first place where the face values are different. 3. Compare these digits. The digit with the greatest face value in this place represents the greater of two numbers. Example: 0.3450 0.1474 Also, remember to line up the decimal points and put in zeros as the end of decimals to make the decimal places the same for ease of ordering. 0.34 <was >now add a zero and it becomes 0.340 0.147 0.147
13.2.4. 7-2 Operations On Decimals I. Algorithm for addition and subtraction of terminating decimals, 1. Use base-ten blocks as on page 421. 2. Change it to a problem we already know how to solve such as: 2.16 + 1.73 = (2 + 1/10 + 6/100) + (1 + 7/10 + 3/100) (2 + 1) + (1/10 + 7/10) + (6/100 + 3/100) 3 + 8/10 + 9/100 = 3.89 3. Line up the decimals points and add or subtract: 2.16 or 2.1 1.73 1.73 3.89 3.83
13.2.4.1. II. Algorithm for multiplying decimals 1. 4.62 ∙ 2.4 = 462 ∙ 24 = 462 ∙24 = 462 ∙ 24 = 11,088 = 11.088 100 10 102 ∙ 101 102 ∙101 103 2. Compute the following way: Remember: if there is n digits to the right of the decimal point in one number and m digits to the right of the decimal point in a second number, multiply the two numbers, ignoring the decimals, and then place the decimal point so that there are n + m digits to the right of the decimal point in the product. 1.43 x 6.2 286 858x 8.866
13.2.4.1.1. 3. Scientific Notation- Use when you have either very small or very large numbers. 93,000,000 = 9.3 X 107 Scientific notation, a positive number written as a product of a number greater than or equal to 1 and less than 10 and an integer power of 10. When you have 0.000078 = 7.8 x 10-5 *Note how to do this on your calculator- pg. 426 • Convert to standard numerals: a. 6.84 x 10-5 = b. 3.12 x 107= c. –(4.08 x 104)=
13.3. (7-3) Nonterminating Decimals
13.3.1. Ways to convert some rational numbers to decimals: 7 = 7 = 7∙ 53 = 875 = 0.875 8 23 23∙53 1000 Or use as a division problem- “numerator divided by denominator point zero, zero, zero” “n÷ d . 0 0 0” 7/8 = 7 8.000
13.3.2. Repeating Decimals 2/11 = 0.181818181 The repeating block of digits is the repetend. Using your calculator, find the answer for: a. 1/7 b. 2/13
13.3.3. Writing Repeating Decimals as Fractions See page 443 A short way of writing the repeating decimals 0.323232… is 0.32. You can write this repeating decimals as a fraction. Write 1n = 0.32323232…Since two digits repeat, multiply both sides by 100 to get 100n= 32.323232… 100n = 32.32323232… - 1n = 0.32323232… 99n= 32 n= 32 99
13.3.4. Ordering Repeating Decimals Write the decimals one under the other, in their equivalent forms without the bars, and line up the decimal points (or place values) as follows: 1.34783478 1.34782178